QUATERNIONIC CONTACT EINSTEIN STRUCTURES AND THE QUATERNIONIC CONTACT YAMABE PROBLEM

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1 QUATERNIONIC CONTACT EINSTEIN STRUCTURES AND THE QUATERNIONIC CONTACT YAMABE PROBLEM STEFAN IVANOV, IVAN MINCHEV, AND DIMITER VASSILEV Abstract. A partial solution of the quaternionic contact Yamabe problem on the quaternionic sphere is given. It is shown that the torsion of the Biquard connection vanishes exactly when the trace-free part of the horizontal Ricci tensor of the Biquard connection is zero and this occurs precisely on 3-Sasakian manifolds. All conformal deformations sending the standard flat torsion-free quaternionic contact structure on the quaternionic Heisenberg group to a quaternionic contact structure with vanishing torsion of the Biquard connection are explicitly described. A 3-Hamiltonian form of infinitesimal conformal automorphisms of quaternionic contact structures is presented. Contents 1. Introduction 2 2. Quaternionic contact structures and the Biquard connection 6 3. The torsion and curvature of the Biquard connection The torsion The Curvature Tensor QC-Einstein quaternionic contact structures The Bianchi identities Examples of qc-einstein structures Proof of Theorem Conformal transformations of a qc-structure Conformal transformations preserving the qc-einstein condition Quaternionic Heisenberg group. Proof of Theorem Special functions and pseudo-einstein quaternionic contact structures Quaternionic pluriharmonic functions Quaternionic pluriharmonic functions on hypercomplex manifold Anti-CRF functions on Quaternionic contact manifold Infinitesimal Automorphisms contact manifolds QC vector fields Quaternionic contact Yamabe problem The Divergence Formula 52 Date: March 27, Mathematics Subject Classification. 58G30, 53C17. Key words and phrases. Yamabe equation, quaternionic contact structures, Einstein structures. This project has been funded in part by the National Academy of Sciences under the [Collaboration in Basic Science and Engineering Program 1 Twinning Program] supported by Contract No. INT from the National Science Foundation. The contents of this publication do not necessarily reflect the views or policies of the National Academy of Sciences or the National Science Foundation, nor does mention of trade names, commercial products or organizations imply endorsement by the National Academy of Sciences or the National Science Foundation. 1

2 2 STEFAN IVANOV, IVAN MINCHEV, AND DIMITER VASSILEV 8.2. Partial solutions of the QC-Yamabe problem Proof of Theorem References Introduction The Riemannian [LP] and CR Yamabe problems [JL1, JL2, JL3, JL4] have been a fruitful subject in geometry and analysis. Major steps in the solutions is the understanding of the conformally flat cases. A model for this setting is given by the corresponding spheres, or equivalently, the Heisenberg groups with, respectively, 0-dimensional and 1-dimensional centers. The equivalence is established through the Cayley transform [K], [CDKR1] and [CDKR2], which in the Riemannian case is the usual stereographic projection. In the present paper we consider the Yamabe problem on the quaternionic Heisenberg group (three dimensional center). This problem turns out to be equivalent to the quaternionic contact Yamabe problem on the unit (4n+3)-dimensional sphere in the quaternionic space due to the quaternionic Cayley transform, which is a conformal quaternionic contact transformation (see the proof of Theorem 1.2). The central notion is the quaternionic contact structure (QC structure for short), [Biq1, Biq2], which appears naturally as the conformal boundary at infinity of the quaternionic hyperbolic space, see also [P, GL, FG]. Namely, a QC structure (η, Q) on a (4n+3)-dimensional smooth manifold M is a codimension 3 distribution H, such that, at each point p M the nilpotent Lie algebra H p (T p M/H p ) is isomorphic to the quaternionic Heisenberg algebra H m Im H. This is equivalent to the existence of a 1-form η = (η 1, η 2, η 3 ) with values in R 3, such that, H = Ker η and the three 2-forms dη i H are the fundamental 2-forms of a quaternionic structure Q on H. A special phenomena here, noted by Biquard [Biq1], is that the 3-contact form η determines the quaternionic structure as well as the metric on the horizontal bundle in a unique way. Of crucial importance is the existence of a distinguished linear connection, see [Biq1], preserving the QC structure and its Ricci tensor and scalar curvature Scal, defined in (3.34), and called correspondingly qc-ricci tensor and qc-scalar curvature. The Biquard connection will play a role similar to the Tanaka-Webster connection [We] and [T] in the CR case. The quaternionic contact Yamabe problem, in the considered setting, is about the possibility of finding in the conformal class of a given QC structure one with constant qc-scalar curvature. The question reduces to the solvability of the Yamabe equation (5.8). As usual if we take the conformal factor in a suitable form the gradient terms in (5.8) can be removed and one obtains the more familiar form of the Yamabe equation. In fact, taking the conformal factor of the form η = u 1/(n+1) η reduces (5.8) to the equation Lu 4 n + 2 n + 1 u u Scal = u2 1 Scal, where is the horizontal sub-laplacian, cf. (5.2), and Scal and Scal are the qc-scalar curvatures correspondingly of (M, η) and (M, η), and 2 = 2Q Q 2, with Q = 4n+6. In the case of the quaternionic Heisenberg group, cf. Section 4.1, the equation is n ( Lu T 2 α u + Xαu 2 + Yα 2 u + Zαu 2 ) = n + 1 4(n + 2) u2 1 Scal. α=1 This is also, up to a scaling, the Euler-Lagrange equation of the non-negative extremals in the L 2 Folland-Stein embedding theorem [Fo] and [FSt], see [GV1] and [Va2]. On the other hand, on a

3 QUATERNIONIC CONTACT STRUCTURES AND THE YAMABE PROBLEM 3 compact quaternionic contact manifold M with a fixed conformal class [η] the Yamabe equation characterizes the non-negative extremals of the Yamabe functional defined by Υ(u) = 4 n + 2 M n + 1 u 2 + Scal u 2 dv g, u 2 dv g = 1, u > 0. M When the Yamabe constant λ(m) def = λ(m, [η]) = inf{υ(u) : M u2 dv g = 1, u > 0} is less than that of the sphere the existence of solutions can be constructed with the use of suitable coordinates see [W] and [JL2]. The present paper can be considered as a contribution towards the soluiton of the Yamabe problem in the case when the Yamabe constant of the considered quaternionic contact manifold is equal to the Yamabe constant of the unit sphere with its standard quaternionic contact structure, which is induced from the embedding in the quaternion (n + 1)-dimensional space. It is also natural to conjecture that if the quaternionic contact structure is not locally equivalent to the standard sphere then the Yamabe constant is less than that of the sphere, see [JL4] for a proof in the CR case. The results of the present paper will be instrumental for the analysis of these and some other questions concerning the geometric analysis on quaternionic contact structures. In this article we provide a partial solution of the Yamabe problem on the quaternionic sphere with its standard contact quaternionic structure or, equivalently, the quaternionic Heisenberg group. Note that according to [GV2] the extremals of the above variational problem are C functions, so we will not consider regularity questions in this paper. Furthermore, according to [Va1] or [Va2] the infimum is achieved, and the extremals are solutions of the Yamabe equation. Let us observe that [GV2] solves the same problem in a more general setting, but under the assumption that the solution is invariant under a certain group of rotation. If one is on the flat models, i.e., the groups of Iwasawa type [CDKR1] the assumption in [GV2] is equivalent to the a-priori assumption that, up to a translation, the solution is radial with respect to the variables in the first layer. The proof goes on by using the moving plane method and showing that the solution is radial also in the variables from the center, after which a very non-trivial identity is used to determine all cylindrical solutions. In this paper the a-priori assumption is of a different nature, see further below, and the method has the potential of solving the general problem. The strategy, following the steps of [LP] and [JL3], is to solve the Yamabe problem on the quaternionic sphere by replacing the non-linear Yamabe equation by an appropriate geometrical system of equations which could be solved. Our first observation is that if the qc-ricci tensor is trace-free (qc-einstein condition) then the qc-scalar curvature is constant (Theorem 4.9). Studying conformal deformations of QC structures preserving the qc-einstein condition, we describe explicitly all global functions on the quaternionic Heisenberg group sending conformally the standard flat QC structure to another qc-einstein structure. Our result here is the following Theorem. Theorem 1.1. Let Θ = 1 Θ 2h be a conformal deformation of the standard qc-structure Θ on the quaternionic Heisenberg group G (H). If Θ is also qc-einstein, then up to a left translation the function h is given by [ (1 h = c + ν q 2 ) 2 + ν 2 (x 2 + y 2 + z )] 2, where c and ν are positive constants. All functions h of this form have this property. The crucial observation reducing the Yamabe equation to the system preserving the qc-einstein condition is Proposition 8.2 which asserts that, under some extra conditions, QC structure with constant qc-scalar curvature obtained by a conformal transformation of a qc-einstein structure on compact manifold must be again qc-einstein. The prove of this relies on detailed analysis of the

4 4 STEFAN IVANOV, IVAN MINCHEV, AND DIMITER VASSILEV Bianchi identities for the Biquard connection. Using the quaternionic Cayley transform combined with Theorem 1.1 lead to our second main result. Theorem 1.2. Let η = f η be a conformal deformation of the standard qc-structure η on the quaternionic sphere S 4n+3. Suppose η has constant qc-scalar curvature. a) If n > 1 then any one of the following two conditions i) the vertical space of η is integrable, ii) the function f is the real part of an anti-crf function, implies that up to a multiplicative constant η is obtained from η by a conformal quaternionic contact automorphism. b) If n = 1 and the vertical space of η is integrable then up to a multiplicative constant η is obtained from η by a conformal quaternionic contact automorphism. The definition of conformal quaternionic contact automorpism can be found in Definition 7.6. The solutions (conformal factors) we find agree with those conjectured in [GV1]. It might be possible to dispense of the extra assumption in Theorem 1.2. In a subsequent paper [IMV] we give such a proof for the seven dimensional sphere. Studying the geometry of the Biquard connection, our main geometrical tool towards understanding the geometry of the Yamabe equation, we show that the qc-einstein condition is equivalent to the vanishing of the torsion of Biquard connection. Furthermore, in our third main result we give a local characterization of such spaces. Theorem 1.3. Let (M 4n+3, g, Q) be a QC manifold with non-zero qc scalar curvature Scal 0. The next conditions are equivalent: a) (M 4n+3, g, Q) is qc-einstein manifold; b) M is locally 3-Sasakian in the sense that locally there exists a SO(3)-matrix Ψ with smooth entries, such that, the local QC structure ( 16n(n+2) Scal Ψ η, Q) is 3-Sasakian; c) The torsion of the Biquard connection is identically zero. In particular, a qc-einstein manifold is Einstein manifold with positive Riemannian scalar curvature and if complete it is compact with finite fundamental group. In addition, in Theorem 7.11 we show that the above conditions are equivalent to the property that every Reeb vector field, defined in (2.5), is an infinitesimal generator of a conformal quaternionic contact automorphism, cf. Definition 7.7. In the paper we also develop useful tools necessary for the geometry and analysis on QC manifolds. We define and study some special functions, which will be relevant in the geometric analysis on quaternionic contact and hypercomplex manifolds as well as properties of infinitesimal automorphisms of QC structures. In particular, the considered anti-regular functions will be relevant in the study of qc-pseudo-einstein structures, cf. Definition 6.1. Organization of the paper: In the following two chapters we describe in details the notion of a quaternionic contact manifold, abbreviate sometimes to QC-manifold, and the Biquard connection, which is central to the paper. In Chapter 4 we write explicitly the Bianchi identities and derive a system of equations satisfied by the divergences of some important tensors. As a result we are able to show that qc-einstein manifolds, i.e., manifolds for which the restriction to the horizontal space of the qc-ricci tensor is proportional to the metric, have constant scalar curvature, see Theorem 4.9. The proof uses Theorem 4.8 in which we derive a relation between the horizontal divergences of certain Sp(n)Sp(1)-invariant tensors. By introducing an integrability condition on the horizontal bundle we define hyperhermitian contact structures, see Definition 4.14, and with the help of Theorem 4.8 we prove Theorem 1.3.

5 QUATERNIONIC CONTACT STRUCTURES AND THE YAMABE PROBLEM 5 Chapter 5 describes the conformal transformations preserving the qc-einstein condition. Note that here a conformal quaternionic contact transformation between two quaternionic contact manifold is a diffeomorphism Φ which satisfies Φ η = µ Ψ η, for some positive smooth function µ and some matrix Ψ SO(3) with smooth functions as entries and η = (η 1, η 2, η 3 ) t is considered as an element of R 3. One defines in an obvious manner a point-wise conformal transformation. Let us note that the Biquard connection does not change under rotations as above, i.e., the Biquard connection of Ψ η and η coincides. In particular, when studying conformal transformations we can consider only transformations with Φ η = µ η. We find all conformal transformations preserving the qc-einstein condition on the quaternionis Heisenberg group or, equivalently, on the quaternionic sphere with their standard contact quaternionic structures proving Theorem 1.1. Chapter 6 concerns a special class of functions, which we call anti-regular, defined respectively on the quaternionic space, real hyper-surface in it, or on a quaternionic contact manifold, cf. Definitions 6.6 and 6.15 as functions preserving the quaternionic structure. The anti-regular functions play a role somewhat similar to those played by the CR functions, but the analogy is not complete. The real parts of such functions will be also of interest in connection with conformal transformation preserving the qc-einstein tensor and should be thought of as generalization of pluriharmonic functions. Let us stress explicitly that regular quaternionic functions have been studied extensively, see [S] and many subsequent papers, but they are not as relevant for the considered geometrical structures. Antiregular functions on hyperkähler and quaternionic Kähler manifolds are studied in [CL1, CL2, LZ] in a different context, namely in connection with minimal surfaces and quaternionic maps between quaternionic Kähler manifolds. The notion of hypercomplex contact structures will appear in this section again since on such manifolds the real part of anti-crf functions, see (6.17) for the definition, have some interesting properties, cf. Theorem 6.20 In Chapter 7 we study infinitesimal conformal automorphisms of QC structures (QC-vector fields) and show that they depend on three functions satisfying some differential conditions thus establishing a 3-hamiltonian form of the QC-vector fields (Proposition 7.8). The formula becomes very simple expression on a 3-Sasakian manifolds (Corollary 7.9). We characterize the vanishing of the torsion of Biquard connection in terms of the existence of three vertical vector fields whose flow preserves the metric and the quaternionic structure. Among them, 3-Sasakian manifolds are exactly those admitting three transversal QC-vector fields (Theorem 7.11, Corollary 7.14). In the last section we complete the proof of our main result Theorem 1.2. Notation 1.4. a) Let us note explicitly, that in this paper for a one form θ we use dθ(x, Y ) = Xθ(Y ) Y θ(x) θ([x, Y ]). b) We shall denote with h the horizontal gradient of the function h, see (5.1), while dh means as usual the differential of the function h. c) The triple {i, j, k} will denote a cyclic permutation of {1, 2, 3}, unless it is explicitly stated otherwise. Acknowledgements S.Ivanov is a Senior Associate to the Abdus Salam ICTP. The paper was completed during the visit of S.I. in Max-Plank-Institut für Mathematik, Bonn. S.I. thanks ICTP and MPIM, Bonn for providing the support and an excellent research environment. I.Minchev is a member of the Junior Research Group Special Geometries in Mathematical Physics founded by the Volkswagen Foundation The authors would like to thank The National Academies for the support. It is a pleasure to acknowledge the role of Centre de Recherches Mathématiques and CIRGET, Montréal where the project initiated. The authors also would like to thank University of California, Riverside and University of Sofia for hosting the respective visits of the authors.

6 6 STEFAN IVANOV, IVAN MINCHEV, AND DIMITER VASSILEV 2. Quaternionic contact structures and the Biquard connection The notion of Quaternionic Contact Structure has been introduced by O.Biquard in [Biq1] and [Biq2]. Namely, a quaternionic contact structure (QC structure for short) on a (4n+3)-dimensional smooth manifold M is a codimension 3 distribution H, such that, at each point p M the nilpotent step two Lie algebra H p (T p M/H p ) is isomorphic to the quaternionic Heisenberg algebra H n Im H. The quaternionic Heisenberg algebra structure on H n Im H is obtained by the identification of H n Im H with the algebra of the left invariant vector fields on the quaternionic Heisenberg group, see Section 5.2. In particular, the Lie bracket is given by the formula [(q o, ω o ), (q, ω)] = 2 Im q o q, where q = (q 1, q 2,..., q n ), q o = (qo, 1 qo, 2..., qo n ) H n and ω, ω o Im H with q o q = n α=1 qα o q α, see Section for notations concerning H. It is important to observe that if M has a quaternionic contact structure as above then the definition implies that the distribution H and its commutators generate the tangent space at every point. The following is another, more explicit, definition of a quaternionic contact structure. Definition 2.1. [Biq1] A quaternionic contact structure ( QC-structure) on a 4n + 3 dimensional manifold M, n > 1, is the data of a codimension three distribution H on M equipped with a CSp(n)Sp(1) structure, i.e., we have i) a fixed conformal class [g] of metrics on H; ii) a 2-sphere bundle Q over M of almost complex structures, such that, locally we have Q = {ai 1 + bi 2 +ci 3 : a 2 +b 2 +c 2 = 1}, where the almost complex structures I s : H H, I 2 s = 1, s = 1, 2, 3, satisfy the commutation relations of the imaginary quaternions I 1 I 2 = I 2 I 1 = I 3 ; iii) H is locally the kernel of a 1-form η = (η 1, η 2, η 3 ) with values in R 3 and the following compatibility condition holds (2.1) 2g(I s X, Y ) = dη s (X, Y ), s = 1, 2, 3, X, Y H. A manifold M with a structure as above will be called also quaternionic contact manifold (QC manifold) and denoted by (M, [g], Q). We note that if in some local chart η is another form, with corresponding ḡ [g] and almost complex structures Īs, s = 1, 2, 3, then η = µ Ψ η for some Ψ SO(3) and a positive function µ Typical examples of manifolds with QC-structures are totally umbilical hypersurfaces in quaternionic Kähler or hyperkähler manifold, see Proposition 6.12 for the latter. It is instructive to consider the case when there is a globally defined one-form η. The obstruction to the global existence of η is encoded in the first Pontrjagin class [AK]. Besides clarifying the notion of a QC-manifold, most of the time, for example when considering the Yamabe equation, we shall work with a QC-structure for which we have a fixed globally defined contact form. In this case, if we rotate the R 3 -valued contact form and the almost complex structures by the same rotation we obtain again a contact form, almost complex structures and a metric (the latter is unchanged) satisfying the above conditions. On the other hand, it is important to observe that given a contact form the almost complex structures and the horizontal metric are unique if they exist. Finally, if we are given the horizontal bundle and a metric on it, there exists at most one sphere of associated contact forms with a corresponding sphere Q of almost complex structures [Biq1]. Besides the non-uniqueness due to the action of SO(3), the 1-form η can be changed by a conformal factor, in the sense that if η is a form for which we can find associated almost complex structures and metric g as above, then for any Ψ SO(3) and a positive function µ, the form µ Ψ η also has an associated complex structures and metric. In particular, when µ = 1 we obtain a whole unit sphere of contact forms, and we shall denote, as already mentioned, by Q the corresponding sphere bundle of associated triples of almost complex structures. With the above consideration in mind we introduce the following notation.

7 QUATERNIONIC CONTACT STRUCTURES AND THE YAMABE PROBLEM 7 Notation 2.2. We shall denote with (M, η) a QC-manifold with a fixed globally defined contact form. (M, g, Q) will denote a QC-manifold with a fixed metric g and a sphere bundle of almost complex structures Q. In this case we have in fact a Sp(n)Sp(1) structure, i.e., we are working with a fixed metric on the horizontal space. Correspondingly, we shall denote with η any (locally defined) associated contact form. We recall the definition of the Lie groups Sp(n), Sp(1) and Sp(n)Sp(1). Let us identify H n = R 4n and let H acts on H n by right multiplications, λ(q)(w ) = W q 1. This defines a homomorphism λ : {unit quaternions} SO(4n) with the convention that SO(4n) acts on R 4n on the left. The image is the Lie group Sp(1). Let λ(i) = I 0, λ(j) = J 0, λ(k) = K 0. The Lie algebra of Sp(1) is sp(1) = span{i 0, J 0, K 0 }. The group Sp(n) is Sp(n) = {O SO(4n) : OB = BO for all B Sp(1)} or Sp(n) = {O M n (H) : O Ōt = I}, and O Sp(n) acts by (q 1, q 2,..., q n ) t O (q 1, q 2,..., q n ) t. Denote by Sp(n)Sp(1) the product of the two groups in SO(4n). Abstractly, Sp(n)Sp(1) = (Sp(n) Sp(1))/Z 2. The Lie algebra of the group Sp(n)Sp(1) is sp(n) sp(1). Any endomorphism Ψ of H can be decomposed with respect to the quaternionic structure (Q, g) uniquely into four Sp(n)-invariant parts Ψ = Ψ +++ +Ψ + +Ψ + +Ψ +, where Ψ +++ commutes with all three I i, Ψ + commutes with I 1 and anti-commutes with the others two and etc. Explicitly, 4Ψ +++ = Ψ I 1 ΨI 1 I 2 ΨI 2 I 3 ΨI 3, 4Ψ + = Ψ I 1 ΨI 1 + I 2 ΨI 2 + I 3 ΨI 3, 4Ψ + = Ψ + I 1 ΨI 1 I 2 ΨI 2 + I 3 ΨI 3, 4Ψ + = Ψ + I 1 ΨI 1 + I 2 ΨI 2 I 3 ΨI 3. The two Sp(n)Sp(1)-invariant components are given by (2.2) Ψ [3] = Ψ +++, Ψ [ 1] = Ψ + + Ψ + + Ψ +. Denoting the corresponding (0,2) tensor via g by the same letter one sees that the Sp(n)Sp(1)- invariant components are the projections on the eigenspaces of the Casimir operator (2.3) = I 1 I 1 + I 2 I 2 + I 3 I 3 corresponding, respectively, to the eigenvalues 3 and 1, see [CSal]. If n = 1 then the space of symmetric endomorphisms commuting with all I i, i = 1, 2, 3 is 1-dimensional, i.e. the [3]-component of any symmetric endomorphism Ψ on H is proportional to the identity, Ψ 3 = Ψ 2 4 Id H. There exists a canonical connection compatible with a given quaternionic contact structure. This connection was discovered by O. Biquard [Biq1] when the dimension (4n+3) > 7 and by D. Duchemin [D] in the 7-dimensional case. The next result due to O. Biquard is crucial in the quaternionic contact geometry. Theorem 2.3. [Biq1] Let (M, g, Q) be a quaternionic contact manifold of dimension 4n + 3 > 7 and a fixed metric g on H in the conformal class [g]. Then there exists a unique connection with torsion T on M 4n+3 and a unique supplementary subspace V to H in T M, such that: i) preserves the decomposition H V and the metric g; ii) for X, Y H, one has T (X, Y ) = [X, Y ] V ; iii) preserves the Sp(n)Sp(1)-structure on H, i.e., g = 0 and Q Q; iv) for ξ V, the endomorphism T (ξ,.) H of H lies in (sp(n) sp(1)) gl(4n); v) the connection on V is induced by the natural identification ϕ of V with the subspace sp(1) of the endomorphisms of H, i.e. ϕ = 0. In (iv) the inner product on End(H) is given by (2.4) g(a, B) = tr(b A) = g(a(e a ), B(e a )),

8 8 STEFAN IVANOV, IVAN MINCHEV, AND DIMITER VASSILEV where A, B End(H), {e 1,..., e 4n } is some g-orthonormal basis of H. We shall call the above connection the Biquard connection. Biquard [Biq1] also described the supplementary subspace V explicitly, namely, locally V is generated by vector fields {ξ 1, ξ 2, ξ 3 }, such that η s (ξ k ) = δ sk, (ξ s dη s ) H = 0, (2.5) (ξ s dη k ) H = (ξ k dη s ) H, s, k {1, 2, 3}. The vector fields ξ 1, ξ 2, ξ 3 are called Reeb vector fields or fundamental vector fields. If the dimension of M is seven, the conditions (2.5) do not always hold. Duchemin shows in [D] that if we assume, in addition, the existence of Reeb vector fields as in (2.5), then Theorem 2.3 holds. Henceforth, by a QC structure in dimension 7 we shall always mean a QC structure satisfying (2.5). Notice that equations (2.5) are invariant under the natural SO(3) action. Using the Reeb vector fields we extend g to a metric on M by span{ξ 1, ξ 2, ξ 3 } = V H and g(ξ s, ξ k ) = δ sk. The extended metric does not depend on the action of SO(3) on V, but it changes in an obvious manner if η is multiplied by a conformal factor. Clearly, the Biquard connection preserves the extended metric on T M, g = 0. We shall also extend the quternionic structure by setting I s V = 0. Suppose the Reeb vector fields {ξ 1, ξ 2, ξ 3 } have been fixed. The restriction of the torsion of the Biquard connection to the vertical space V satisfies [Biq1] (2.6) T (ξ i, ξ j ) = λξ k [ξ i, ξ j ] H, where λ is a smooth function on M. Let us recall that here and further in the paper the indices follow the convention 1.4. Further properties of the Biquard connection are encoded in the properties of the torsion endomorphism T ξ = T (ξ,.) : H H, ξ V. Decomposing the endomorphism T ξ (sp(n) + sp(1)) into symmetric part T 0 ξ and skew-symmetric part b ξ, T ξ = T 0 ξ + b ξ, we summarize the description of the torsion due to O. Biquard in the following Proposition. Proposition 2.4. [Biq1] The torsion T ξ is completely trace-free, (2.7) trt ξ = g(t ξ (e a ), e a ) = 0, trt ξ I = where e 1... e 4n is an orthonormal basis of H. The symmetric part of the torsion has the properties: (2.8) (2.9) T 0 ξ i I i = I i T 0 ξ i, i = 1, 2, 3; g(t ξ (e a ), Ie a ) = 0, I Q, I 2 (T 0 ξ 2 ) + = I 1 (T 0 ξ 1 ) +, I 3 (T 0 ξ 3 ) + = I 2 (T 0 ξ 2 ) +, I 1 (T 0 ξ 1 ) + = I 3 (T 0 ξ 3 ) +. The skew-symmetric part can be represented in the following way (2.10) b ξi = I i u, i = 1, 2, 3, where u is a traceless symmetric (1,1)-tensor on H which commutes with I 1, I 2, I 3. If n = 1 then the tensor u vanishes identically, u = 0 and the torsion is a symmetric tensor, T ξ = T 0 ξ. 3. The torsion and curvature of the Biquard connection Let (M 4n+3, g, Q) be a quaternionic contact structure on a 4n + 3-dimensional smooth manifold. Working in a local chart, we fix the vertical space V = span{ξ 1, ξ 2, ξ 3 } by requiring the conditions (2.5). The fundamental 2-forms ω i, i = 1, 2, 3 of the quaternionic structure Q are defined by (3.1) 2ω i H = dη i H, ξ ω i = 0, ξ V.

9 QUATERNIONIC CONTACT STRUCTURES AND THE YAMABE PROBLEM 9 Define three 2-forms θ i, i = 1, 2, 3 by the formulas (3.2) θ i = 1 2 {d((ξ j dη k ) H ) + (ξ i dη j ) (ξ i dη k )} H = 1 2 {d(ξ j dη k ) + (ξ i dη j ) (ξ i dη k )} H dη k (ξ j, ξ k )ω k + dη k (ξ i, ξ j )ω i. Define, further, the corresponding (1, 1) tensors A i by g(a i (X), Y ) = θ i (X, Y ), X, Y H The torsion. Due to (3.1), the torsion restricted to H has the form (3.3) T (X, Y ) = [X, Y ] V = 2 ω s (X, Y )ξ s, X, Y H. The next two Lemmas provide some useful technical facts. Lemma 3.1. Let D be any differentiation of the tensor algebra of H. Then we have D(I i ) I i = I i D(I i ), i = 1, 2, 3, I 1 D(I 1 ) + = I 2 D(I 2 ) +, I 1 D(I 1 ) + = I 2 D(I 2 ) +, I 1 D(I 1 ) + = I 2 D(I 2 ) +. Proof. The proof is a straightforward consequence of the next identities 0 = I 2 (D(I 1 ) I 2 D(I 1 )I 2 ) + I 1 (D(I 2 ) I 1 D(I 2 )I 1 ) = I 2 D(I 1 ) + + I 1 D(I 2 ) +, 0 = D( Id V ) = D(I i I i ) = D(I i )I i + I i D(I i ). With L denoting the Lie derivative, we set L = L H. Lemma 3.2. The following identities hold true. (3.4) L ξ 1 I 1 = 2Tξ 0 1 I 1 + dη 1 (ξ 1, ξ 2 )I 2 + dη 1 (ξ 1, ξ 3 )I 3, (3.5) L ξ 1 I 2 = 2Tξ I2 2I 3 ũ + dη 1 (ξ 2, ξ 1 )I (dη 1(ξ 2, ξ 3 ) dη 2 (ξ 3, ξ 1 ) dη 3 (ξ 1, ξ 2 ))I 3 (3.6) L ξ 2 I 1 = 2T 0 ξ 2 + I1 + 2I 3 ũ + dη 2 (ξ 1, ξ 2 )I ( dη 1(ξ 2, ξ 3 ) + dη 2 (ξ 3, ξ 1 ) dη 3 (ξ 1, ξ 2 ))I 3, where the symmetric endomorphism ũ on H, commuting with I 1, I 2, I 3, is defined by (3.7) 2ũ = I 3 ((L ξ 1 I 2 ) + ) (dη 1(ξ 2, ξ 3 ) dη 2 (ξ 3, ξ 1 ) dη 3 (ξ 1, ξ 2 ))Id H In addition, we have six more identities, which can be obtained with a cyclic permutation of (1,2,3). Proof. For all k, l = 1, 2, 3 we have (3.8) L ξk ω l (X, Y ) = L ξk g(i l X, Y ) + g((l ξk I l )X, Y ) Cartan s formula yields (3.9) L ξk ω l = ξ k (dω l ) + d(ξ k ω l ). A direct calculation using (3.1) gives (3.10) 2ω l = (dη l ) H = dη l η s (ξ s dη l ) + dη l (ξ s, ξ t )η s η t. 1 s<t 3

10 10 STEFAN IVANOV, IVAN MINCHEV, AND DIMITER VASSILEV Combining (3.10) and (3.9) we obtain, after a short calculation, the following identities (3.11) (3.12) (L ξ1 ω 1 ) H = (dη 1 (ξ 1, ξ 2 )ω 2 + dη 1 (ξ 1, ξ 3 )ω 3 ) H 2(L ξi ω j ) H = (d(ξ i dη j ) (ξ i dη k ) (ξ k dη j )) H, where i j k i, i, j, k {1, 2, 3}. Clearly, (3.11) and (3.8) imply (3.4). Furthermore, using (2.5) and (3.12) twice for i = 1, j = 2 and i = 2, j = 1, we find (3.13) (L ξ1 ω 2 + L ξ2 ω 1 ) H = 1 2 (d(ξ 1 dη 2 ) + d(ξ 2 dη 1 )) H On the other hand, (3.8) implies (3.14) 2T 0 ξ 1 I 2 + L ξ 1 I 2 + 2T 0 ξ 2 I 1 + L ξ 2 I 1 Let us decompose (3.14) into Sp(n)-invariant components: (3.15) (3.16) = dη 1 (ξ 2, ξ 1 )ω 1 + dη 2 (ξ 1, ξ 2 )ω 2 + (dη 1 (ξ 2, ξ 3 ) + dη 2 (ξ 1, ξ 3 ))ω 3. = dη 1 (ξ 2, ξ 1 )I 1 + dη 2 (ξ 1, ξ 2 )I 2 + (dη 1 (ξ 2, ξ 3 ) + dη 2 (ξ 1, ξ 3 ))I 3. (L ξ 1 I 2 ) + = 2T 0 ξ 1 + I2 + dη 1 (ξ 2, ξ 1 )I 1, (L ξ 2 I 1 ) + = 2T 0 ξ 2 + I1 + dη 2 (ξ 1, ξ 2 )I 2, Using (3.16) and (3.7), we obtain (L ξ 1 I 2 + L ξ 2 I 1 ) + = (dη 1 (ξ 2, ξ 3 ) + dη 2 (ξ 1, ξ 3 ))I 3. 2ũ = I 3 ((L ξ 2 I 1 ) + ) ( dη 1(ξ 2, ξ 3 ) + dη 2 (ξ 3, ξ 1 ) dη 3 (ξ 1, ξ 2 ))Id H. The latter, together with (3.7), tells us that ũ commutes with all I Q. Now, Lemma 3.1 with D = L implies (3.5) and (3.6). The vanishing of the symmetric part of the left hand side in (3.8) for k = 1, l = 2, combined with (3.18) and (3.5) yields 0 = 2g(I 3 ũx, Y ) 2g(I 3 ũy, X). As ũ commutes with all I Q we conclude that ũ is symmetric. The rest of the identities can be obtained through a cyclic permutation of (1,2,3). We describe the properties of the quaternionic contact torsion more precisely in the next Proposition. Proposition 3.3. The torsion of the Biquard connection satisfies the identities: (3.17) (3.18) T ξi = T 0 ξ i + I i u, i = 1, 2, 3, T 0 ξ i = 1 2 L ξ i g, i = 1, 2.3, (3.19) u = ũ tr(ũ) 4n Id H, where the symmetric endomorphism ũ on H commuting with I 1, I 2, I 3 satisfies ũ = 1 2 I 1A ( dη1 (ξ 2, ξ 3 ) + dη 2 (ξ 3, ξ 1 ) + dη 3 (ξ 1, ξ 2 ) ) Id H (3.20) = 1 2 I 2A ( dη1 (ξ 2, ξ 3 ) dη 2 (ξ 3, ξ 1 ) + dη 3 (ξ 1, ξ 2 ) ) Id H = 1 2 I 3A ( dη1 (ξ 2, ξ 3 ) + dη 2 (ξ 3, ξ 1 ) dη 3 (ξ 1, ξ 2 ) ) Id H. For n = 1 the tensor u = 0 and ũ = tr(ũ) 4 Id H.

11 QUATERNIONIC CONTACT STRUCTURES AND THE YAMABE PROBLEM 11 Proof. Expressing the Lie derivative in terms of the Biquard connection, using that preserves the splitting H V, we see that for X, Y H we have L ξi g(x, Y ) = g( X ξ i, Y ) + g( Y ξ i, X) + g(t ξi X, Y ) + g(t ξi Y, X) = 2g(T 0 ξ i X, Y ). To show that ũ satisfies (3.20), insert (3.12) into (3.2) to get (3.21) θ 3 = (L ξ1 ω 2 ) H dη 2 (ξ 1, ξ 2 )ω 3 + dη 2 (ξ 3, ξ 1 )ω 3. A substitution of (3.8) and (3.5) in (3.21) gives (3.22) A 3 = 2T 0 ξ 1 + I2 2I 3 ũ + dη 1 (ξ 2, ξ 1 )I 1 dη 2 (ξ 1, ξ 2 )I (dη 1(ξ 2, ξ 3 ) + dη 2 (ξ 3, ξ 1 ) dη 3 (ξ 1, ξ 2 ))I 3. Now, by comparing the (+++) component on both sides of (3.22) we see the last equality of (3.20). The rest of the identities can be obtained with a cyclic permutation of (1,2,3). Turning to the rest of the identities, let Σ 2 and Λ 2 denote, respectively, the subspaces of symmetric and skew-symmetric endomorphisms of H. Let skew : End(H) Λ 2 be the natural projection with kernel Σ 2. We have 4[T ξi ] (Σ2 sp(n)) = 3skew(T ξ i ) + I 1 skew(t ξi )I 1 + I 2 skew(t ξi )I 2 + I 3 skew(t ξi )I 3 = (skew(t ξi ) + I s skew(t ξi )I s ). According to Theorem 2.3, T ξ X H for X H, ξ V. Hence, (3.23) T (ξ, X) = ξ X [ξ, X] H = ξ X L ξ(x). An application of (3.23) gives (3.24) g(4[t ξi ] (Σ2 sp(n)) X, Y ) = g ( ( ξi I s )X, I s Y ) {g ( (L ξi I s )X, I s Y ) g((l ξi I s )Y, I s X)}. Let B(H) be the orthogonal complement of Σ 2 sp(n) sp(1) in End(H). Obviously, B(H) Λ 2 and we have the following splitting of End(H) into mutually orthogonal components (3.25) End(H) = Σ 2 sp(n) sp(1) B(H). If Ψ is an arbitrary section of the bundle Λ 2 of M, the orthogonal projection of Ψ into B(H) is given by [Ψ] B(H) = Ψ + + Ψ + + Ψ + [Ψ] sp(1), where [Ψ] sp(1) is the orthogonal projection of Ψ onto sp(1). We have also [Ψ] sp(1) = 1 3 4n 4n g(ψe a, I s e a )I s. Theorem (iv) and the splitting (3.25) yield (3.26) T ξi = [T ξi ] (sp(n) sp(1)) = [T ξi ] Σ 2 + [T ξi ] B(H) = T 0 ξ i + [T ξi ] (Σ 2 sp(n)) [T ξ i ] sp(1). Using (3.24), Lemma 3.2 and the fact that I s ( ξi I s ) sp(1), we compute 4[T ξi ] (Σ2 sp(n)) [T { } ξ i ] sp(1) = skew(is (L ξ i I s )) [I s (L ξ i I s )] sp(1) (3.27) = skew(2i s Tξ 0 i I s ) + 4u = 4u.

12 12 STEFAN IVANOV, IVAN MINCHEV, AND DIMITER VASSILEV Inserting (3.27) in (3.26) completes the proof. The Sp(n)-invariant splitting of (3.22) leads to the following Corollary. Corollary 3.4. The (1,1)-tensors A i satisfy the equalities A = 2T 0 ξ 1 + I2, A + 3 = dη 1 (ξ 2, ξ 1 )I 1, A + 3 = dη 2 (ξ 1, ξ 2 )I 2, A + 3 = 2I 3 ũ (dη 1(ξ 2, ξ 3 ) + dη 2 (ξ 3, ξ 1 ) dη 3 (ξ 1, ξ 2 ))I 3. Analogous formulas for A 1 and A 2 can be obtained by a cyclic permutation of (1, 2, 3). Proposition 3.5. The covariant derivative of the quaternionic contact structure with respect to the Biquard connection is given by (3.28) I i = α j I k + α k I j, where the sp(1)-connection 1-forms α s are determined by (3.29) α i (X) = dη k (ξ j, X) = dη j (ξ k, X), X H, ξ i V ; ( tr(ũ) α i (ξ s ) = dη s (ξ j, ξ k ) δ is 2n + 1 ) (3.30) 2 (dη 1(ξ 2, ξ 3 ) + dη 2 (ξ 3, ξ 1 ) + dη 3 (ξ 1, ξ 2 )), s = 1, 2, 3. Proof. The equality (3.29) is proved by Biquard in [Biq1]. Using (3.23), we obtain ξs I i = [T ξs, I i ] + L ξ s I i = [T 0 ξ s, I i ] + u[i s, I i ] + L ξ s I i. An application of Lemma 3.2 completes the proof. Corollary 3.6. The covariant derivative of the distribution V is given by ξ i = α j ξ k + α k ξ j. We finish this section by expressing the Biquard connection in terms of the Levi-Civita connection D g of the metric g, namely, we have (3.31) B Y = D g B Y + {((D g B η s)y )ξ s + η s (B)(I s u I s )Y }, B T M, Y H. Indeed, for B = X H formula (3.31) follows from the equation X Y = [D g X Y ] H. If B V we may assume B = ξ 1 and for Z H we compute 2g(D g ξ 1 Y, Z) = ξ 1 g(y, Z) + g([ξ 1, Y ], Z) g([ξ 1, Z], Y ) g([y, Z], ξ 1 ) = = (L ξ1 g)(y, Z) + 2g([ξ 1, Y ], Z) + dη 1 (Y, Z) = 2g(T ξ1 Y + [ξ 1, Y ], Z) 2g(I 1 uy, Z) + 2g(I 1 Y, Z) = 2g( ξ1 Y, Z) 2g((I 1 u I 1 )Y, Z). In the above calculation we used (3.23) and Proposition 3.3. Note that the covariant derivatives B ξ s are also determined by (3.31) in view of the relation g( B ξ s, ξ k ) = 1 4n g( BI s, I k ), s, k = 1, 2, The Curvature Tensor. Let R = [, ] [, ] be the curvature tensor of. For any B, C Γ(T M) the curvature operator R BC preserves the QC structure on M since preserves it. In particular R BC preserves the distributions H and V, the quaternionic structure Q on H and the (2, 1) tensor ϕ. Moreover, the action of R BC on V is completely determined by its action on H, R BC ξ i = ϕ 1 ([R BC, I i ]), i = 1, 2, 3. Thus, we may regard R BC as an endomorphism of H and we have R BC sp(n) sp(1).

13 QUATERNIONIC CONTACT STRUCTURES AND THE YAMABE PROBLEM 13 Definition 3.7. The Ricci 2-forms ρ i are defined by ρ i (B, C) = 1 4n g(r(b, C)e a, I i e a ), B, C Γ(T M). Hereafter {e 1,..., e 4n } will denote an orthonormal basis of H. We decompose the curvature into sp(n) sp(1)-parts. Let RBC 0 sp(n) denote the sp(n)-component. Lemma 3.8. The curvature of the Biquard connection decomposes as follows (3.32) (3.33) R BC = R 0 BC + ρ 1 (B, C)I 1 + ρ 2 (B, C)I 2 + ρ 3 (B, C)I 3. [R BC, I i ] = 2( ρ j (B, C)I k + ρ k (B, C)I j ), B, C Γ(T M), ρ i = 1 2 (dα i + α j α k ), where the connection 1-forms α s are determined in (3.29), (3.30). Proof. The first two identities follow directly from the definitions. Using (3.28), we calculate [R BC, I 1 ] = B (α 3 (C)I 2 α 2 (C)I 3 ) C (α 3 (B)I 2 α 2 (B)I 3 ) (α 3 ([B, C])I 2 α 2 ([B, C])I 3 ) = (dα 2 + α 3 α 1 )(B, C)I 3 + (dα 3 + α 1 α 2 )(B, C)I 2. Now (3.32) completes the proof. Definition 3.9. The quaternionic contact Ricci tensor ( qc-ricci tensor for short) and the qc-scalar curvature Scal of the Biquard connection are defined by (3.34) Ric(B, C) = g(r(e a, B)C, e a ), Scal = Ric(e a, e a ). It is known, cf. [Biq1], that the qc-ricci tensor restricted to H is symmetric. In addition, we define six Ricci-type tensors ζ i, τ i, i = 1, 2, 3 as follows (3.35) ζ i (B, C) = 1 4n g(r(e a, B)C, I i e a ), τ i (B, C) = 1 4n g(r(e a, I i e a )B, C). We shall show that all Ricci-type contractions evaluated on the horizontal space H are determined by the components of the torsion. First, define the following 2-tensors on H (3.36) T 0 (X, Y ) def = g((t 0 ξ 1 I 1 + T 0 ξ 2 I 2 + T 0 ξ 3 I 3 )X, Y ), U(X, Y ) def = g(ux, Y ), X, Y H. Lemma The tensors T 0 and U are Sp(n)Sp(1)-invariant trace-free symmetric tensors with the properties: (3.37) (3.38) T 0 (X, Y ) + T 0 (I 1 X, I 1 Y ) + T 0 (I 2 X, I 2 Y ) + T 0 (I 3 X, I 3 Y ) = 0, 3U(X, Y ) U(I 1 X, I 1 Y ) U(I 2 X, I 2 Y ) U(I 3 X, I 3 Y ) = 0. Proof. The lemma follows directly from (2.8), (2.10) of Proposition 2.4. We turn to a Lemma, which shall be used later.

14 14 STEFAN IVANOV, IVAN MINCHEV, AND DIMITER VASSILEV Lemma For any X, Y H, B H V, we have (3.39) (3.40) Ric(B, I i Y ) + 4nζ i (B, Y ) = 2ρ j (B, I k Y ) 2ρ k (B, I j Y ), ζ i (X, Y ) = 1 2 ρ i(x, Y ) + 1 2n g(i iux, Y ) + 2n 1 2n g(t 0 ξ i X, Y ) The Ricci 2-forms evaluated on H satisfy (3.41) + 1 2n g(i jt 0 ξ k X, Y ) 1 2n g(i kt 0 ξ j X, Y ). ρ 1 (X, Y ) = 2g(T 0 + ξ 2 I 3 X, Y ) 2g(I 1 ux, Y ) tr(ũ) n ω 1(X, Y ), ρ 2 (X, Y ) = 2g(T 0+ ξ 3 I 1 X, Y ) 2g(I 2 ux, Y ) tr(ũ) n ω 2(X, Y ), ρ 3 (X, Y ) = 2g(T 0 + ξ 1 I 2 X, Y ) 2g(I 3 ux, Y ) tr(ũ) n ω 3(X, Y ). The 2-forms τ s evaluated on H satisfy τ 1 (X, Y ) = ρ 1 (X, Y ) + 2g(I 1 ux, Y ) g(tξ n 2 I 3 X, Y ), (3.42) τ 2 (X, Y ) = ρ 2 (X, Y ) + 2g(I 2 X, Y ) g(tξ n 3 I 1 X, Y ), τ 3 (X, Y ) = ρ 3 (X, Y ) + 2g(I 3 X, Y ) g(tξ n 1 I 2 X, Y ). For n = 1 the above formulas hold with U = 0. Proof. From (3.32) we have Ric(B, I 1 Y ) + 4nζ 1 (B, Y ) = = {R(e a, B, I 1 Y, e a ) + R(e a, B, Y, I 1 e a )} { 2ρ 2 (e a, B)ω 3 (Y, e a ) + 2ρ 3 (e a, B)ω 2 (Y, e a ))} = 2ρ 2 (B, I 3 Y ) 2ρ 3 (B, I 2 Y ), Use (3.2) and (3.33) to get ρ 1 (X, Y ) = A 1 (X, Y ) 1 2 α 1([X, Y ] V ) = A 1 (X, Y )+ 3 ω s(x, Y )α 1 (ξ s ). Now, Corollary 3.4 and Corollary 3.5 imply the first equality in (3.41). The other two equalities in (3.41) can be obtained in the same manner. Letting b(x, Y, Z, W ) = 2σ X,Y,Z { 3 l=1 ω l(x, Y )g(t ξl Z, W )}, where σ X,Y,Z is the cyclic sum over X, Y, Z, we have (3.43) (3.44) b(x, Y, e a, I 1 e a ) = 4g(I 1 ux, Y ) + 8g(I 2 T 0 + ξ 3 X, Y ), b(e a, I 1 e a, X, Y ) = (8n 4)g(T 0 ξ 1 X, Y ) + (8n + 4)g(I 1 ux, Y ) + 4g(T 0 ξ 2 I 3 X, Y ) 4g(T 0 ξ 3 I 2 X, Y ).

15 QUATERNIONIC CONTACT STRUCTURES AND THE YAMABE PROBLEM 15 The first Bianchi identity gives (3.45) 4n(τ 1 (X, Y ) + 2ζ 1 (X, Y )) = {R(e a, I 1 e a, X, Y ) + R(X, e a, I 1 e a, Y ) + R(I 1 e a, X, e a, Y )} = b(e a, I 1 e a, X, Y ), (3.46) 4n(τ 1 (X, Y ) ρ 1 (X, Y )) = = 1 2 {R(e a, I 1 e a, X, Y ) R(X, Y, e a, I 1 e a )} {b(e a, I 1 e a, X, Y ) b(e a, I 1 e a, Y, X) b(e a, X, Y, I 1 e a ) + b(i 1 e a, X, Y, e a )}. Consequently, (3.43), (3.44), (3.45) and (3.46) yield the first set of equalities in (3.42) and (3.40). The other equalities in (3.42) and (3.40) can be shown similarly. Theorem Let (M 4n+3, g, Q) be a quaternionic contact (4n + 3)-dimensional manifold, n > 1. For any X, Y H the qc-ricci tensor and the qc-scalar curvature satisfy (3.47) Ric(X, Y ) = (2n + 2)T 0 (X, Y ) + (4n + 10)U(X, Y ) + (2n + 4) tr(ũ) n g(x, Y ) Scal = (8n + 16)tr(ũ). For n = 1, Ric(X, Y ) = 4T 0 (X, Y ) + 6 tr(ũ) n g(x, Y ). Proof. The proof follows from Lemma 3.11, (3.40), (3.41) and (3.39). If n = 1, recall that U = 0 to obtain the last equality. Corollary The qc-scalar curvature satisfies the equalities Scal 2(n + 2) = ρ i (I i e a, e a ) = τ i (I i e a, e a ) = 2 We determine the unknown function λ in (2.6) in the next Corollary. ζ i (I i e a, e a ), i = 1, 2, 3. Corollary The torsion of the Biquard connection restricted to V satisfies the equality (3.48) T (ξ i, ξ j ) = Scal 8n(n + 2) ξ k [ξ i, ξ j ] H. Proof. A small calculation using Corollary 3.6 and Proposition 3.5, gives T (ξ i, ξ j ) = ξi ξ j ξj ξ i [ξ i, ξ j ] = tr(ũ) n ξ k [ξ i, ξ j ] H. Now, the assertion follows from the second equality in (3.47). Corollary The tensors T 0, U, ũ do not depend on the choice of the local basis.

16 16 STEFAN IVANOV, IVAN MINCHEV, AND DIMITER VASSILEV 4. QC-Einstein quaternionic contact structures The aim of this section is to show that the vanishing of the torsion of the quaternionic contact structure implies that the qc-scalar curvature is constant and to prove our classification Theorem 1.3. The Bianchi identities will have an important role in the analysis. Definition 4.1. A quaternionic contact structure is qc-einstein if the qc-ricci tensor is trace-free, Ric(X, Y ) = Scal g(x, Y ), X, Y H. 4n Proposition 4.2. A quaternionic contact manifold (M, g, Q) is a qc-einstein if and only if the quaternionic contact torsion vanishes identically, T ξ = 0, ξ V. Proof. If (η, Q) is qc-einstein structure then T 0 = U = 0 because of (3.47). We will use the same symbol T 0 for the corresponding endomorphism of the 2-tensor T 0 on H. According to (3.36), we have T 0 = T 0 ξ 1 I 1 + T 0 ξ 2 I 2 + T 0 ξ 3 I 3. Using first (2.8) and then (2.9), we compute (4.1) (T 0 ) + = (T 0 ξ 2 ) + I 2 + (T 0 ξ 3 ) + I 3 = 2(T 0 ξ 2 ) + I 2. Hence, T ξ2 = T 0 ξ 2 + I 2 u vanishes. Similarly T ξ1 = T ξ3 = 0. The converse follows from (3.47). Proposition 4.3. For X V and any cyclic permutation (i, j, k) of (1, 2, 3) we have (4.2) (4.3) (4.4) ρ i (X, ξ i ) = X(Scal) 32n(n + 2) (ω i([ξ j, ξ k ], X) ω j ([ξ k, ξ i ], X) ω k ([ξ i, ξ j ], X)), ρ i (X, ξ j ) = ω j ([ξ j, ξ k ], X), ρ i (X, ξ k ) = ω k ([ξ j, ξ k ], X), ρ i (I k X, ξ j ) = ρ i (I j X, ξ k ) = g(t (ξ j, ξ k ), I i X) = ω i ([ξ j, ξ k ], X). Proof. Since preserves the splitting H V, the first Bianchi identity, (3.48) and (3.32) imply (4.5) 2ρ i (X, ξ i ) + 2ρ j (X, ξ j ) = g(r(x, ξ i )ξ j, ξ k ) + g(r(ξ j, X)ξ i, ξ k ) = σ ξi,ξ j,x{g(( ξi T )(ξ j, X), ξ k ) + g(t (T (ξ i, ξ j ), X), ξ k )} = g(( X T )(ξ i, ξ j ), ξ k ) + g(t (T (ξ i, ξ j ), X), ξ k ) = X(Scal) 8n(n + 2) 2ω k([ξ i, ξ j ], X). Summing up the first two equalities in (4.5) and subtracting the third one, we obtain (4.2). Similarly, 2ρ k (ξ j, X) = g(r(ξ j, X)ξ i, ξ j ) = σ ξi,ξ j,x{g(( ξi T )(ξ j, X), ξ j ) + g(t (T (ξ i, ξ j ), X), ξ j )} = g(t (T (ξ i, ξ j ), X), ξ j ) = g(t ( [ξ i, ξ j ] H, X), ξ j ) = g([[ξ i, ξ j ] H, X], ξ j ) = dη j ([ξ i, ξ j ] H, X) = 2ω j ([ξ i, ξ j ], X). Hence, the second equality in (4.3) follows. Analogous calculations show the validity of the first equality in (4.3). Then, (4.4) is a consequence of (4.3) and (3.48). The vertical derivative of the qc-scalar curvature is determined in the next Proposition. Proposition 4.4. On a QC manifold we have (4.6) ρ i (ξ i, ξ j ) + ρ k (ξ k, ξ j ) = 1 16n(n + 2) ξ j(scal). Proof. Since preserves the splitting H V, the first Bianchi identity and (3.48) imply 2(ρ i (ξ i, ξ j ) + ρ k (ξ k, ξ j )) = g(σ ξi,ξ j,ξ k {R(ξ i, ξ j )ξ k }, ξ j ) 1 = g(σ ξi,ξ j,ξ k {( ξi T )(ξ j, ξ k ) + T (T (ξ i, ξ j ), ξ k )}, ξ j ) = 8n(n + 2) ξ j(scal)

17 QUATERNIONIC CONTACT STRUCTURES AND THE YAMABE PROBLEM The Bianchi identities. In order to derive the essential information contained in the Bianchi identities we need the next Lemma, which is an application of a standard result in differential geometry. Lemma 4.5. In a neighborhood of any point p M 4n+3 and an Q-orthonormal basis {X 1 (p), X 2 (p) = I 1 X 1 (p)..., X 4n (p) = I 3 X 4n 3 (p), ξ 1 (p), ξ 2 (p), ξ 3 (p)} of the tangential space at p there exists a Q - orthonormal frame field {X 1, X 2 = I 1 X 1,..., X 4n = I 3 X 4n 3, ξ 1, ξ 2, ξ 3 }, X a p = X a (p), ξ s p = ξ s (p), a = 1,..., 4n, i = 1, 2, 3, such that the connection 1-forms of the Biquard connection are all zero at the point p: (4.7) ( Xa X b ) p = ( ξi X b ) p = ( Xa ξ t ) p = ( ξt ξ s ) p = 0, for a, b = 1,..., 4n, s, t, r = 1, 2, 3. In particular, (( Xa I s )X b ) p = (( Xa I s )ξ t ) p = (( ξt I s )X b ) p = (( ξt I s )ξ r ) p = 0. Proof. Since preserves the splitting H V we can apply the standard arguments for the existence of a normal frame with respect to a metric connection (see e.g. [Wu]). We sketch the proof for completeness. Let { X 1,..., X 4n, ξ 1, ξ 2, ξ 3 } be a Q-orthonormal basis around p such that X a p = X a (p), ξi p = ξ i (p). We want to find a modified frame X a = o b a X b, ξ i = o j i ξ j, which satisfies the normality conditions of the lemma. Let ϖ be the sp(n) sp(1)-valued connection 1-forms with respect to { X 1,..., X 4n, ξ 1, ξ 2, ξ 3 }, X b = ϖ c b X c, ξ s = ϖ t s ξ t, B { X 1,..., X 4n, ξ 1, ξ 2, ξ 3 }. Let {x 1,..., x 4n+3 } be a coordinate system around p such that x a (p) = X a(p), One can easily check that the matrices ( o b a = exp 4n+3 c=1 x 4n+t (p) = ξ t(p), a = 1,..., 4n, t = 1, 2, 3. ) ϖa( b x c ) px c Sp(n), o s t = exp ( 4n+3 c=1 ) ϖt s ( x c ) px c Sp(1) are the desired matrices making the identities (4.7) true. Now, the last identity in the lemma is a consequence of the fact that the choice of the orthonormal basis of V does not depend on the action of SO(3) on V combined with Corollary 3.6 and Proposition 3.5. Definition 4.6. We refer to the orthonormal frame constructed in Lemma 4.5 as a qc-normal frame. Let us fix a qc-normal frame {e 1,..., e 4n, ξ 1, ξ 2, ξ 3 }. We shall denote with X, Y, Z horizontal vector fields X, Y, Z H and keep the notation for the torsion of type (0,3) T (B, C, D) = g(t (B, C), D), B, C, D H V.

18 18 STEFAN IVANOV, IVAN MINCHEV, AND DIMITER VASSILEV Proposition 4.7. On a quaternionic contact manifold (M 4n+3, g, Q) the following identities hold (4.8) (4.9) (4.10) (4.11) 2 ( ea Ric)(e a, X) X(Scal) = 4 Ric(ξ r, I r X) 8n r=1 Ric(ξ s, I s X) = 2ρ q (I t X, ξ s ) + 2ρ t (I s X, ξ q ) + 4n(ρ s (X, ξ s ) ζ s (ξ s, X)) = 2ρ q (I t X, ξ s ) + 2ρ t (I s X, ξ q ) ζ s (ξ s, X) = 1 4n ( ea T )(ξ s, I s X, e a ), ρ r (ξ r, X); r=1 ( ea T )(ξ s, I s X, e a ); where s {1, 2, 3} is fixed and (s, t, q) is an even permutation of (1, 2, 3). Proof. The second Bianchi identity implies 2 ( ea Ric)(e a, X) X(Scal) + 2 Ric(T (e a, X), e a ) + a,b=1 ( ea T )(ξ s, X, I s e a ); R(T (e b, e a ), X, e b, e a ) = 0. Apply (3.3) in the last equality to get (4.8). The first Bianchi identity combined with (2.7), (3.3) and the fact that preserves the orthogonal splitting H V yield ( ) Ric(ξ s, I s X) = ( ea T )(ξ s, I s X, e a ) + 2 ω r (I s X, e a )T (ξ r, ξ s, e a ) = = r=1 ( ea T )(ξ s, I s X, e a ) + 2T (ξ s, ξ t, I q X) + 2T (ξ q, ξ s, I t X), which, together with (4.4), completes the proof of (4.9). In a similar fashion, from the first Bianchi identity, (2.7), (3.3) and the fact that preserves the orthogonal splitting H V we can obtain the proof of (4.10). Finally, take (3.39) with B = ξ i and combine the result with (4.9) to get (4.11). The following Theorem gives relations between Sp(n)Sp(1)-invariant tensors and is crucial for the solution of the Yamabe problem, which we shall undertake in the last Section. We define the horizontal divergence P of a (0,2)-tensor field P with respect to Biquard connection to be the (0,1)-tensor defined by P (.) = 4n ( e a P )(e a,.), where e a, a = 1,..., 4n is an orthonormal basis on H. Theorem 4.8. The horizontal divergences of the curvature and torsion tensors satisfy the system B b = 0, where b = ( n 1 16n(n+2) 0 B = n (n+2) 0, T 0, ) t 3 U, A, d Scal H, j=1 Ric (ξ j, I j. ),

19 QUATERNIONIC CONTACT STRUCTURES AND THE YAMABE PROBLEM 19 with T 0 and U defined in (3.36) and A(X) = g(i 1 [ξ 2, ξ 3 ] + I 2 [ξ 3, ξ 1 ] + I 3 [ξ 1, ξ 2 ], X). Proof. Throughout the proof of Theorem 4.8 (s, t, q) will denote an even permutation of (1, 2, 3). Equations (4.2) and (4.4) yield (4.12) 3 ρ r (X, ξ r ) = 32n(n + 2) X(Scal) 1 2 A(X), (4.13) r=1 ρ q (I t X, ξ s ) = A(X). Using the properties of the torsion described in Proposition 3.3 and (2.8), we obtain (4.14) ( ea T )(ξ s, I s X, e a ) = T 0 (X) 3 U(X), (4.15) ( ea T )(ξ s, X, I s e a ) = T 0 (X) + 3 U(X). Substituting (4.13) and (4.14) in the sum of (4.9) written for s = 1, 2, 3, we obtain the third row of the system. The second row can be obtained by inserting (4.11) into (4.10), taking the sum over s = 1, 2, 3 and applying (4.12), (4.13), (4.14), (4.15). The second Bianchi identity and applications of (3.3) give ( ) [ ] ( ea Ric)(I s X, I s e a ) + 4n( ea ζ s )(I s X, e a ) 2Ric(ξ s, I s X) + 8nζ s (ξ s, X) (4.16) + 8n [ ] ζ s (ξ t, I q X) ζ s (ξ q, I t X) ρ s (ξ t, I q X) + ρ s (ξ q, I t X) = 0. Using (3.39), (3.41) as well as (2.8),(2.9) and (4.1) we obtain the next sequence of equalities [ ] [ ] Ric(I s X, I s e a ) + 4nζ s (I s X, e a ) = 2 ρ s (I q X, I t e a ) ρ s (I t X, I q e a ) (4.17) = = 4T 0 (X, e a ) + 24U(X, e a ) + 3 Scal 2n(n + 2) g(x, e a), [ ] 8n ζ s (ξ t, I q X) ζ s (ξ q, I t X) [ ] 4Ric(ξ s, I s X) 8ρ s (ξ s, X) + 4ρ s (ξ t, I q X) 4ρ s (ξ q, I t X. [ ] 2Ric(ξ s, I s X) + 8nζ s (ξ s, X) = [ ] 4Ric(ξ s, I s X) 4ρ s (ξ t, I q X) + 4ρ s (ξ q, I t X), Substitute (4.17) in (4.16), and then use (4.12) and (4.13) to get the first row of the system. We are ready to prove one of our main observations.